I have cleaned up a few flooded messes and it is nasty and not fun, but I don't think the hazard of cleaning it up is anything near as bad as living with it in your basement or your ducts. Rubber gloves, mask, eyewear to make sure nothing gets in your eyes, coverall or suitable work clothes you can wash or throw out if you really get it on you. Contractor bags, gumption, and just go to it.
If you can bag and get rid of all the damaged stuff in 2-3 hours take that as step one, and enough for one day. Have a hot shower, clean clothes and a drink to congratulate yourself.
Day two go in and clean the building surfaces with vinegar. or borax. Again I would just do one morning or one afternoon at a time. No need to make it a marathon exposure.
There is no special training needed for those jobs, only common sense and taking normal precautions. The occupational hazard may be serious if doing it day in and day out, but much less if you are doing it only once! and no more than a few hours at a time before you get outdoors and breathe some clean air.
It sounds like you need to run a dehumidifier too.
If the place is still wet after you remove all the junk then you need to find where the water is coming from and fix that to get the place dry for long term. This is the kind of savvy that gives the professionals their edge. And they will get in and clean ductwork crawl spaces etc. if it needs to be done.
But even if you manage step 1 you are ahead of the game. Cardboard boxes of junk are the worst thing IMO to be lurking in a flooded space. They host the mold and dampness like they were made for it.
The longer you leave it, the worse it will get.
Of course if you have the money to pay the pros, why not. It's the job no one wants to do.